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When I encounter these two, I usually don't have a problem understanding their meaning in the given context.
Still, I don't really know what makes them different from each other xD
-なくて still is relatively easy for me to determine, since its the negated て form.
"Not liking that person, I go meet him/her."
I can't really determine what ないで is though, and therefore the difference between the two isn't completely clear to me.
"I wanted to study abroad, but I was told by my mother that she wishes that I don't go."
So I do see that it is something along the lines of negation, but I still can't thoroughly classify it's exact form and function :D